Philosophy of Mathematics Jeremy Avigad Final version will appear in Boundas, Constantin editor, The Edinburgh Companion to the 20th Century Philosophies, Edinburgh University Press. The annotations in the bibliography will not appear in the published version. 1. Introduction The philosophy of mathematics plays an important role in analytic philosophy, both as a subject of inquiry in its own right, and as an important landmark in the broader philosophical landscape. Mathematical knowledge has long been regarded as a paradigm of human knowledge with truths that are both necessary and certain, so giving an account of mathematical knowledge is an important part of epistemology. Mathematical objects like numbers and sets are archetypical examples of abstracta, since we treat such objects in our discourse as though they are independent of time and space; finding a place for such objects in a broader framework of thought is a central task of ontology, or metaphysics. The rigor and precision of mathematical language depends on the fact that it is based on a limited vocabulary and very structured grammar, and semantic accounts of mathematical discourse often serve as a starting point for the philosophy of language. Although mathematical thought has exhibited a strong degree of stability through history, the practice has also evolved over time, and some developments have evoked controversy and debate; clarifying the basic goals of the practice and the methods that are appropriate to it is therefore an important foundational and methodological task, locating the philosophy of mathematics within the broader philosophy of science. In this chapter, I will try to convey a modern philosophical understanding of the subject as it practiced today. Our contemporary understanding has been shaped by traditional questions and concerns about the nature of mathematics, and Section 2 provides a broad overview of the general array of philosophical positions in place by the turn of the twentieth century. On the other hand, the nineteenth century is generally taken to represent the birth of ‘modern’ mathematical thought, and many new issues arose with the dramatic conceptual shifts that took place. Sections 3 and 4 summarize some of these developments. Our contemporary philosophical understanding has also been informed by nineteenth and twentieth century developments in mathematical logic, which can be viewed as a reflective, mathematical study of the methods of mathematical reasoning itself. Section 5 tries to explain what we have learned from such a study, in philosophical terms. Finally, Sections 6 and 7 try to convey, in broad strokes, some of the central lines of thought in the philosophy of mathematics today. 2. Traditional questions Traditionally, the two central questions for the philosophy of mathematics are: What are mathematical objects? How do we (or can we) have knowledge of them? Plato offers the following simple answers: abstract mathematical objects, like triangles and spheres, are forms, which have imperfect reflections in this world. Before we are born, our souls have direct interactions with these forms, though we forget most of what we know during the traumatic circumstances of our birth. Recapturing this knowledge is thus a process of recollection, which can be encouraged by the dialectical process. This position is illustrated by Plato's portrayal of Socrates in Meno, when Socrates calls over a slave boy and, through a sequence of questions, brings the boy to understand a simple geometric theorem.1 Socrates leads Meno to the conclusion that since Socrates did not tell the boy the theorem, the boy must have had knowledge of it all along. Although we today may have difficulty with the theory's reliance on otherworldly forms and our soul's prenatal activities, the account does have its advantages: it does justice to the mysterious abstract nature of mathematical objects, and explains why we do not have to appeal to our physical experiences to justify mathematical statements. In contrast, the Aristotelian account of mathematical knowledge holds that mathematical objects, like triangles and spheres, are abstractions from our experiences.2 That is, from our interactions with various roughly spherical objects, we form the concept of a perfect sphere. Reasoning about spheres in general boils down to reasoning about specific spheres we have encountered, qua their sphericity; that is, we deliberately ignore features like size, weight, and material in the discourse. It is this disciplined behaviour that ensures that our conclusions are appropriately general, and even though the spheres we encounter in our experience are not perfect spheres, our conclusions apply insofar as they approximate the latter. Thus the divide between Plato and Aristotle is an early example of tensions between philosophical theories that give primacy to abstract concepts, and those that give primacy to experience. This has formed the basis for the common distinction between rationalists and empiricists among the early modern philosophers, the former taking mathematics and ‘innate ideas’ as the paradigm of knowledge, and the latter basing their accounts of knowledge on the empirical sciences. At times, it can be hard to tell exactly what is at stake. For example, John Locke, as distinctly an empiricist as they come, allows for innate faculties like comparing, compounding, and abstracting, and reflecting on the inner workings of our mind;3 and he agrees that mathematical knowledge consists of certain knowledge of ideas, although these ideas must ultimately spring from experience.4 René Descartes, the prototypical rationalist, maintains that mathematics is the paradigm of knowledge, since its truths can be obtained by a clear and unclouded mind reflecting on clear and distinct ideas. At the end of the Meditations, he allows that 1 Meno, 82-87. Translation by G. M. A. Grube in Cahn (2002). 2 Metaphysics, Book M, 1077b-1078b. Translated by Julia Annas (1976), Aristotle’s Metaphysics: M and N. Oxford: Clarendon Press. 3 Essay Concerning Human Understanding, Book II, Chapter XI. Excerpts in Cahn (2002). 4 Ibid, Book IV, Chapter IV, §6. our physical models must have something to do with the world, since God is not a malicious deceiver. But what we have certain knowledge of are the mathematical concepts and relationships to each other, in contrast to the approximate knowledge that the world conforms, more or less, to our models. The distinctions between mathematical and scientific knowledge are expressed in different ways by the early modern philosophers, and such reformulations provide different insights. For example, Gottfried Leibniz distinguished between necessary and contingent truths: the former, including the truths of mathematics, are true in all possible worlds, and could not be otherwise; the latter, like the facts of scientific discovery, could have been different.5 Similarly, David Hume distinguished between relations of ideas and matters of fact. The truths of mathematics are grouped under relations of ideas, ‘every affirmation of which is either intuitively or demonstratively certain.’ In contrast, any matter of fact could have been otherwise: ‘the contrary of every matter of fact is still possible, because it can never imply a contradiction and is conceived by the mind with the same facility and distinctness, as if every so conformable to reality.’6 The work of Immanuel Kant went a long way to clarify and defuse some of the differences between rationalist and empiricist stances. For example, Kant helpfully underscored the difference between asserting that a concept arises from experience, and asserting that it arises with experience.7 According to Kant, the issue is not whether we are born with a concept of triangle or whether we develop this concept over time. Rather, the relevant fact is that an appropriate justification for assertions about triangles need not make reference to experience. Truths that have this character, like the truths of mathematics, Kant called a priori. The remaining truths, that is, those we justify by referring to our experiences, he called a posteriori. Kant went on to observe that one can separately distinguish between judgements that depend only on the definition of the concepts involved, and those that don't. In fact, Kant only considered assertions in subject-predicate form; he called a judgement ‘A is B’ analytic when ‘the predicate B belongs to the subject A as something that is (covertly) contained in this concept A.’8 For example, the statement that triangles have three sides relies only on the knowledge of the definition of triangle. This notion of analyticity requires clarification, and we will return to it in Section 4. A judgement that is not analytic he called synthetic. According to Kant, any statement that is analytic is a priori; if the truth of a statement rests only on the definition of the concepts involved, then one need not appeal to experience to justify it. Conversely, any statement that is a posteriori has to be synthetic. But is there a middle ground, consisting of statements that are a priori but synthetic? Kant argued that nontrivial truths of mathematics fall exactly into this category. For example, the justification for the fact that 5 + 7 = 12 cannot be found in the definition of 5 Discourse on Metaphysics, §13. Translation by Roger Ariew and Daniel Garber in Cahn (2002). 6 An Enquiry Concerning Human Understanding, Section IV, Part I. Excerpts in Cahn (2002). 7 Critique of Pure Reason, B1. Translation by Werner S. Pluhar in Cahn (2002). 8 This quote and the views that follow can be found in Critique of Pure Reason, B10-B21. the concept of 5, or that of 7, or that of 12, or that of ‘+’, or that of ‘=‘. But, on the other hand, we don't appeal to experimentation to justify the assertion. Thus, this statement is a priori but synthetic. Given that there are a priori synthetic truths, the central question for Kant is to explain how this is possible.